Characteristic differential equation of a T-junction: diffusive shear work exchange from its head loss coefficients

Negative head loss coefficients in branched junctions, has been controversial for long. Herwig et al. showed, that the cause is a diffusive shear work exchange. Based on their work, a new junction internal model is described, while the conventional head loss is named external. The latter is obtained experimentally, while the first cannot. This fact seems to push back reaching a practical solution. Conventionally two head `loss' coefficients are required. However the internal model needs three: two `pure' head loss coefficients and a work coefficient. Based on previous works, the paper shows that the missing equation comes from the Minimum Energy Dissipation Principle (MinEDP). The characteristic differential equation of a T-junction is discovered, which relates the two `pure' head loss coefficients. A particular case is presented to illustrate how to obtain the internal model from the external one by using it. The results are applied to empirical and numerical data from the same branched junction; Zhu's measurements (external) and Herwig's CFD computations (internal), respectively. Finally, an example illustrates how powerful this new method is to analyse a new, recently published, exhaust return duct.